CONDITIONAL IDENTITIES TRIGONOMETRY

Trigonometric identities are true for all admissible values of the angle involved. There are some trigonometric identities which satisfy the given additional conditions. Such identities are called conditional trigonometric identities.

Here we are going to see some examples to show how to solve problems on conditional trigonometric identities.

Examples

Example 1 :

If A + B + C = π/2, prove that

sin2A + sin2B + sin2C = 4cosAcosBcosC

Solution :

sin2A + sin2B + sin2C :

= 2sin(A + B)cos(A - B) + sin2C

= 2sin(90 - C)cos(A - B) + 2sinCcosC

= 2cosCcos(A - B) + 2sinCcosC

= 2cosC[cos(A - B) + sinC]

= 2cosC[cos(A - B) + sin(90 - (A + B)]

= 2cosC[cos(A - B) + cos(A + B)]

= 2cosC[2cosAcos(-B)]

= 2cosC[2cosAcosB]

= 4cosAcosBcosC

Hence proved.

Example 2 :

If A + B + C = π/2, prove that

cos2A + cos2B + cos2C = 1 + 4sinAsinB cosC

Solution :

cos2A + cos2B + cos2C :

Let us use the formula of (cosC + cosD) for cos2A + cos2B

= 2cos(A + B)cos(A - B) + cos2C

= 2cos(90 - C)cos(A - B) + 1 - 2sin2C

= 2sinCcos(A - B) + 1 - 2sin2C

= 1 + 2sinC[cos(A - B) - sinC]

= 1 + 2sinC[cos(A - B) - sin(90 - (A + B)]

= 1 + 2sinC[cos(A - B) - cos(A + B)]

= 1 + 2sinC[-2sinAsin(-B)]

= 1 + 2sinC[2sinAsinB]

= 1 + 4sinAsinBsinC

Hence proved.

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